Many statistical procedures estimate how an outcome is affected by factors that may influence it. For example, a multivariate statistical model may represent variations of a dependent variable as a function of a set of independent variables. A limitation of these procedures is that they may not be able to completely resolve joint effects among two or more independent variables.
A “joint effect” is an effect that is the joint result of two or more factors. “Statistical joint effects” are those joint effects remaining after the application of statistical methods. “Cooperative resolution” is the application of cooperative game theory to resolve statistical joint effects.
A “performance measure” is a statistic derived from a statistical model that describes some relevant aspect of that model such as its quality or the properties of one of its variables. A performance measure may be related to a general consideration such as assessing the accuracy of a statistical model's predictions. Cooperative resolution can completely attribute the statistical model's performance, as reflected in a performance measure, to an underlying source such as the statistical model's independent variables.
Most performance measures fall in to one of two broad categories. The first category of performance measure gauges an overall “explanatory power” of a model. The explanatory power of a model is closely related to its accuracy. A typical measure of explanatory power is a percentage of variance of a dependent variable explained by a multivariate statistical model.
The second category of performance measure gauges a “total effect.” Measures of total effect address the magnitude and direction of effects. An example of such a total effect measure is a predicted value of a dependent variable in a multivariate statistical model.
Some of the limits of the prior art with respect to the attribution of explanatory power and total effects may be illustrated with reference to a standard multivariate statistical model. A multivariate statistical model is commonly used to determine a mathematical relationship between its dependent and independent variables. One common measure of explanatory power is a model's “R2” coefficient. This coefficient takes on values between zero percent and 100% in linear statistical models, a common statistical model. An R2 of a model is a percentage of a variance of a dependent variable, i.e., a measure of its variation, explained by the model. The larger an R2 value, the better the model describes a dependent variable.
The explanatory power of a multivariate statistical model is an example of a statistical joint effect. As is known in the art, in studies based on a single independent variable, it is common to report the percentage of variance explained by that variable. An example from the field of financial economics is E. Fama and K. French, “Common risk factors in the returns on stocks and bonds,” Journal of Financial Economics, v. 33, n. 1. 1993, pp. 3-56. In multivariate statistical models, however, it may be difficult or impossible, relying only on the existing statistical arts, to isolate a total contribution of each independent variable.
The total effect of a multivariate statistical model in its estimation of a dependent variable is reflected in estimated coefficients for its independent variables. If there are no interaction variables, independent variables that represent joint variation of two or more other independent variables, then, under typical assumptions, it is possible to decompose this total effect into separate effects of the independent variables. However, in the presence of interaction variables there is no accepted method in the art for resolving the effects of the interaction variables to their component independent variables.
One principal accepted method to determine the explanatory power of independent variables in a multivariate statistical model is by assessment of their “statistical significance.” An independent variable is statistically significant if a “significance test” determines that its true value is different than zero. As is known in the art, a significance test has a “confidence level.” If a variable is statistically significant at the 95% confidence level, there is a 95% chance that its true value is not zero. An independent variable is not considered to have a “significant effect” on the dependent variable unless it is found to be statistically significant. Independent variables may be meaningfully ranked by their statistical significance. However, this ranking may provide limited insight into their relative contributions to explained variance.
Cooperative game theory can be used to resolve statistical joint effects problems. As is known in the art, “game theory” is a mathematical approach to the study of strategic interaction among people. Participants in these games are called “players.” Cooperative game theory allows players to make contracts and has been used to solve problems of bargaining over the allocation of joint costs and benefits. A “coalition” is a group of players that have signed a binding cooperation agreement. A coalition may also comprise a single player.
A cooperative game is defined by assigning a “worth,” i.e., a number, to each coalition in the game. The worth of a coalition describes how much it is capable of achieving if its players agree to act together. Joint effects in a cooperative game are reflected in the worths of coalitions in the game. In a cooperative game without joint effects, the worth of any coalition would be the sum of the worths of the individual players in the coalition.
There are many methods available to determine how the benefits of cooperation among all players should be distributed among the players. (Further information on cooperative game theory can be found in Chapter 9 of R. G. Myerson, Game Theory: Analysis of Conflict, Cambridge: Harvard University Press, 1992, pp. 417-482, which is incorporated by reference.) Cooperative game theory has long been proposed as a method to allocate joint costs or benefits among a group of players. In most theoretical work the actual joint costs or benefits are of an abstract nature. The practical aspects of using of cooperative game theory to allocate joint costs has received somewhat more attention. See, for example, H. P. Young, ed., Cost Allocation: Methods, Principles, Applications, New York: North Holland, 1985.
Techniques from the prior art typically cannot be used to satisfactorily resolve statistical joint effects in cooperative games. Thus, it is desirable to use cooperative game theory to resolve statistical joint effects problems.
There have been attempts in the prior art to decompose joint explanatory power. For example, R. H. Lindeman, P. F. Merenda, and R. Z. Gold, in Introduction to Bivariate and Multivariate Analysis, 1980, Scott, Foresman, and Company, Glenview, Ill., ISBN 0-673-15099-2, pp. 119-127, describe a method of variance decomposition based on averaging the marginal contribution of a variable to R2 over all possible orderings of variables. The authors discuss a method that generates the Shapley value of a variable in a statistical cooperative game using R2 as a measure of explanatory power. W. Kruskal, in “Concepts of relative importance,” The American Statistician, 1987, v. 41, n. 1, pp 6-10, and A. Chevan and M. Sutherland, in “Hierarchical partitioning,” The American Statistician, 1991, v. 45, n. 2, 90-96, describe related methods based on the marginal contributions over all possible orderings of variables.
Also, it is known in the art that the explained variance in a regression can be decomposed into linear components. The variance assigned to an independent variable i in this decomposition is the sum over all variables j of the expression βiσijβj, where βj is the regression coefficient associated with a variable j and σij is the covariance between independent variables i and j. This decomposition corresponds to the Shapley value of a statistical cooperative game using explained variance as a performance measure and using coefficients the complete statistical model to determine the worths of all coalitions.
Statistical cooperative games based on total effects may have coalitions with negative worths. It may be desirable to use proportional allocation principles in resolving these joint effects, however the proportional value cannot be applied to cooperative games with negative worths. It is desirable to demonstrate how proportional allocation effects determined in a first cooperative control game may be applied in a second cooperative allocation game that has negative coalitional worths through the use of an intergrated proportional control value of a controlled allocation game.
Statistical cooperative games may have large numbers of players. The calculation of value functions for large games can use large quantities of computer time. M. Conklin and S. Lipovetsky, in “Modem marketing research combinatorial computations: Shapley value versus TURF tools,” 1998 S-Plus User Conference, disclose a method for approximating the Shapley and weighted Shapley values. It is desirable to approximate the powerpoint, the proportional value, and integrated proportional control values. It also desirable to show how the precision of value approximations may be ascertained.